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Clean integral 2 × 2 matrices are characterized. Up to similarity the strongly clean matrices are completely determined and large classes of uniquely clean matrices are found. In particular, classes of uniquely clean matrices which are not strongly clean are found.

This paper is devoted to study the following Schrödinger-Poisson system
$$\{\begin{array}{l}-\Delta u+\left(\lambda a\left(x\right)+b\left(x\right)\right)u+K\left(x\right)\varphi u=f\left(u\right),x\in {\mathbb{R}}^{3},\\ -\Delta \varphi =4\pi K\left(x\right){u}^{2},x\in {\mathbb{R}}^{3},\end{array}$$
where λ is a positive parameter, *a* ∈ *C*(R^{3},R^{+}) has a bounded potential well Ω = *a*
^{−1}(0), *b* ∈ *C*(R^{3}, R) is allowed to be sign-changing, *K* ∈ *C*(R^{3}, R^{+}) and *f* ∈ *C*(R, R). Without the monotonicity of *f*(*t*)=/|*t*|^{3} and the Ambrosetti-Rabinowitz type condition, we establish the existence and exponential decay of positive multi-bump solutions of the above system for $$\lambda \ge \overline{\Lambda}$$, and obtain the concentration of a family of solutions as λ →+∞, where $$\overline{\Lambda}>0$$ is determined by terms of *a*, *b*, *K* and *f*. Our results improve and generalize the ones obtained by C. O. Alves, M. B. Yang [3] and X. Zhang, S. W. Ma [38].

Let *R* be a ring with an endomorphism *σ* and *F* ∪ {0} the free monoid generated by *U* = {*u*
_{1}, ..., *u _{t}*} with 0 added, and

*M*=

*F*∪ {0}/(

*I*) where

*I*is the set of certain monomial in

*U*such that

*M*

^{n}= 0, for some

*n*. Then we can form the non-semiprime skew monoid ring

*R*[

*M*; σ]. An element

*a*∈

*R*is uniquely strongly clean if

*a*has a unique expression as

*a*=

*e*+

*u*, where

*e*is an idempotent and

*u*is a unit with

*ea*=

*ae*. We show that a σ-compatible ring

*R*is uniquely clean if and only if

*R*[

*M*; σ] is a uniquely clean ring. If

*R*is strongly π-regular and uniquely strongly clean, then

*R*[

*M*; σ] is uniquely strongly clean. It is also shown that idempotents of

*R*[

*M*; σ] (and hence the ring

*R*[

*x*; σ]=(

*x*

^{n})) are conjugate to idempotents of

*R*and we apply this to show that

*R*[

*M*; σ] over a projective-free ring

*R*is projective-free. It is also proved that if

*R*is semi-abelian and σ(

*e*) =

*e*for each idempotent

*e*∈

*R*, then

*R*[

*M*; σ] is a semi-abelian ring.

We exhibit a general family of distributions named “Kumaraswamy odd Burr G family of distributions” with four additional parameters to generalize any existing baseline distribution. Some statistical properties of the family are derived, including *r*
^{th} moments, *m*
^{th} incomplete moments, moment generating function and entropies. The parameters of the family are estimated by the maximum likelihood (ML) method for complete sam- ples as well as censored samples. Some sub-models of the family are considered and it is noted that their density functions can be symmetric, left-skewed, right-skewed, unimodal, bimodal and their hazard rate functions can be increasing, decreasing, bathtub, upside- down bathtub and J-shaped. Simulation is carried out for one of the sub-models to check the asymptotic behavior of the ML estimates. Applications to reliability (complete and censored) data are carried out to check the usefulness of some sub-models of the family.

We study point processes associated with coupon collector’s problem, that are defined as follows. We draw with replacement from the set of the first *n* positive integers until all elements are sampled, assuming that all elements have equal probability of being drawn. The point process we are interested in is determined by ordinal numbers of drawing elements that didn’t appear before. The set of real numbers is considered as the state space. We prove that the point process obtained after a suitable linear transformation of the state space converges weakly to the limiting Poisson random measure whose mean measure is determined.

We also consider rates of convergence in certain limit theorems for the problem of collecting pairs.

The class *CR* of completely regular semigroups considered as algebras with binary multiplication and unary operation of inversion forms a variety. Kernel, trace, local and core relations, denoted by **K**, **T**, **L** and **C**, respectively, are quite useful in studying the structure of the lattice *L*(*CR*) of subvarieties of *CR*. They are equivalence relations whose classes are intervals. Their ends are used for defining operators on *L*(*CR*).

Starting with a few band varieties, we repeatedly apply operators induced by upper ends of classes of these relations and characterize corresponding classes up to certain variety low in the lattice *L*(*CR*). We consider only varieties whose origin are “central” band varieties, that is those in the middle column of the lattice *L*(*B*) of band varieties. Several diagrams represent the (semi)lattices studied.

We prove that if *I*
_{k} are disjoint blocks of positive integers and *n*
_{k} are independent random variables on some probability space (Ω,*F*,P) such that *n*
_{k} is uniformly distributed on *I*
_{k}, then
$${N}^{-1/2}{\displaystyle \sum _{k=1}^{N}\left(\mathrm{sin}2\pi {n}_{k}x-E\left(\mathrm{sin}2\pi {n}_{k}x\right)\right)}$$
has, with P-probability 1, a mixed Gaussian limit distribution relative to the probability space ((0, 1),*B*, λ), where *B* is the Borel *σ*-algebra and λ is the Lebesgue measure. We also investigate the case when *n*
_{k} have continuous uniform distribution on disjoint intervals *I*
_{k} on the positive axis.

Let *G* be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂*G*. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes ε^{p),θ(G,ω)} and $${\epsilon}^{p),\theta \left({G}^{-},\omega \right)}$$, 1 < p < ∞, in the term of the *r*th, *r* = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.

In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller’s characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the Lévy-Wiener theorem. This is a special case of an open problem which is proposed by Sato (2014), Chaumont and Yor (2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter λ in the DPCP class cannot be arbitrarily small.

Let *R* be a ring. The purpose of this paper is to study the existence and the representation for the anti-triangular matrix $$\left[\begin{array}{cc}a& b\\ c& 0\end{array}\right]$$ under some conditions, where *a*, *b*, *c* ∈ *R*. The results extend recent works given in the literature.